Question

A square matrix with non-negative elements and unit row sums is called as a:

• A. Stochastic matrix
• B. Positive definite matrix
• C. Unitary matrix
• D. All of these

Hint:

Non negative entries with each row summing to 1 is precisely the textbook definition of one particular named matrix.

Answer 1

The Correct Option is A

Step 1: Recall the definition. A square matrix \(P=(p_{ij})\) is called a stochastic matrix if every entry \(p_{ij} \geq 0\) and every row sums to 1, that is \(\sum_j p_{ij}=1\) for each \(i\).

Step 2: A positive definite matrix is defined through the sign of the quadratic form \(x^T A x > 0\) for every nonzero \(x\), which has nothing to do with row sums or entry signs.

Step 3: A unitary matrix \(U\) satisfies \(U U^{*} = I\), again a completely different condition unrelated to non negativity or row sums.

Step 4: The stated description, non negative entries with unit row sums, matches exactly and only the definition of a stochastic matrix.

Answer: option (A).